Cross Section (Geometry)

# Cross section (geometry)

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In geometry, a cross-section is the intersection of a figure in 2-dimensional space with a line, or of a body in 3-dimensional space with a plane, etc. More plainly, when cutting an object into slices one gets many parallel cross-sections.

Cavalieri's principle states that solids with corresponding cross-sections of equal areas have equal volumes.

The cross-sectional area ([itex]A'[/itex]) of an object when viewed from a particular angle is the total area of the orthographic projection of the object from that angle. For example, a cylinder of height h and radius r has [itex]A' = pi r^2[/itex] when viewed along its central axis, and [itex]A' = 2 pi rh[/itex] when viewed from an orthogonal direction. A sphere of radius r has [itex]A' = pi r^2[/itex] when viewed from any angle. More generically, [itex]A'[/itex] can be calculated by evaluating the following surface integral:

[itex] A' = iint limits_mathrm dmathbf cdot mathbf, [/itex]

where [itex]mathbf[/itex] is a unit vector pointing along the viewing direction toward the viewer, [itex]dmathbf[/itex] is a surface element with outward-pointing normal, and the integral is taken only over the top-most surface, that part of the surface that is "visible" from the perspective of the viewer. For a convex body, each ray through the object from the viewer's perspective crosses just two surfaces. For such objects, the...

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